The man in question’s may have had a Ligamen case, if his first spouse had been previously married.
The man in question may have had a Lack of Form case if his first spouse was a Catholic who did not marry in Catholic form or receive a dispensation from form.
It is also possible that his prior marriage was a natural marriage if he and/or his first spouse were unbaptized. Therefore, it is possible the first marriage was dissolved by either the Pauline or Petrine Privilege.
So, again, many possibilities.
What you can be assured of is that prior to marriage in the Catholic Church a thorough premarital investigation is conducted, and by whatever means was appropriate in this case the diocese determined that he was free to marry your cousin.